No, not even in the 60s, would Christopher become part of the crew. Not in-universe or out.

Before serving on a starship, other than as some sort of advisor, I would think that he would have to graduate Starfleet Academy or some Starfleet OCS at minimum, his USAF rank nonwithstanding. He would also have about three hundred years of catching up to do on the hardware and technology alone.

In the event that he couldn't be sent back to the 1960's, I could see him being a civilian instructor at the Academy, his expertise being 20th century aviation or history, for example, or maybe a successful author or lecturer.

__________________This device isn’t a spaceship, it’s a time machine. It goes backwards, and forwards… it takes us to a place where we ache to go again.